Question: Solve for $x$ and $y$ using elimination. $\begin{align*}-2x+6y &= -6 \\ -x+8y &= 2\end{align*}$
Solution: We can eliminate $x$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-1$ and the bottom equation by $2$ $\begin{align*}2x-6y &= 6\\ -2x+16y &= 4\end{align*}$ Add the top and bottom equations. $10y = 10$ Divide both sides by $10$ and reduce as necessary. $y = 1$ Substitute $1$ for $y$ in the top equation. $-2x+6( 1) = -6$ $-2x+6 = -6$ $-2x = -12$ $x = 6$ The solution is $\enspace x = 6, \enspace y = 1$.